Welfare costs of reclassification risk in the health insurance market

Transcription

1 MPRA Munich Personal RePEc Archive Welfare costs of reclassification risk in the health insurance market Svetlana Pashchenko and Ponpoje Porapakkarm Uppsala University, University of Macau 19. October 211 Online at MPRA Paper No. 4813, posted 5. July 213 4:21 UTC

2 Welfare costs of reclassification risk in the health insurance market Svetlana Pashchenko Ponpoje Porapakkarm Uppsala University University of Macau June 23, 213 Abstract One of the major problems of the U.S. health insurance market is that it leaves individuals exposed to reclassification risk. Reclassification risk arises because the health conditions of individuals evolve over time, while a typical health insurance contract only lasts for one year. A change in the health status can lead to a significant change in the health insurance premium. We study how costly this reclassification risk is for the welfare of consumers. More specifically, we use a general equilibrium model to quantify the implications of introducing guaranteed renewable contracts into the economy calibrated to replicate the key features of the health insurance system in the U.S. Guaranteed renewable contracts are private insurance contracts that can provide protection against reclassification risk even in the absence of consumer commitment or government intervention. We find that the welfare effects from introducing this type of contracts are small implying that the presence of reclassification risk does not impose large welfare losses on consumers. This happens because two institutions in the current U.S. system substitute for the missing explicit contracts that insure reclassification risk: employer-sponsored health insurance and government means-tested transfers. If these two institutional features are removed, the average welfare gains from having access to guaranteed renewable contracts are large and can exceed 2% of the annual consumption. Keywords: health insurance, reclassification risk, dynamic insurance, guaranteed renewable contracts, general equilibrium JEL Classification Codes: D52, D58, D91, G22, I11 We thank all seminar participants at the University of Virginia, Federal Reserve Bank of Richmond, QSPS Summer Workshop, PET Meeting in Bloomington, SED Meeting in Ghent, GRIPS, Asian Meeting of the Econometric Society in Seoul, Uppsala University, and especially, Huberto Ennis, John Jones, and Eric Young, for their comments and suggestions. Porapakkarm acknowledges financial support from the Research and Development Administration Office at the University of Macau. All errors are our own. sap9v@virginia.edu ponpojep@umac.mo 1

3 1 Introduction An important feature of the health insurance market is that a typical insurance policy only lasts for one year while a disease can last for any period of time. This creates the problem of reclassification risk - a risk to face a drastic increase in health premiums when one s health status deteriorates. The fact that standard health insurance contracts leave individuals exposed to reclassification risk is considered an important market failure in the health insurance market (Hendel and Lizzeri, 23; Diamond, 1992). Moreover, the observation that sick people face high health insurance premiums was an important argument for an additional regulation on health insurance market during health reform debate. 1 The goal of this paper is to evaluate how important is the lack of protection against reclassification risk for the welfare of consumers. One way to do this is to compare the current system with the first best solution to the problem of reclassification risk. The first best is to enroll everyone into a long-term health insurance contract. The price of such a contract depends on the average expected medical expenses of all participants. In other words, healthy people make transfers to the sick equalizing the insurance price for all risk categories. These contracts require consumer s commitment because healthy individuals will tend to drop out. As shown by Cochrane (1995), the lack of commitment can be overcome by introducing a special arrangement such as illiquid accounts 2. Another problem with ensuring participation in these contracts is incomplete labor markets. Since premiums are based on average medical expenses but not on individual income, consumers experiencing a sequence of bad income shocks may be unable to pay the premium. This can be solved by introducing income-based transfers. However, since all income redistributive measures have a nontrivial effect on welfare, in the presence of these transfers it is hard to measure a pure welfare effect of reclassification risk. 1 This debate resulted in the major health reform bill that was signed in 21. Among other things, this bill forbids insurance companies to risk-adjust premiums in the individual market. This can be considered as a regulatory way to remove reclassification risk. We discuss implications of our results for the health reform in Section 7. 2 More specifically, Cochrane s idea is to substitute long-term contracts with a sequence of short-term contracts that require consumers who turn out to be healthy to make transfers to insurance firms. Illiquid accounts are needed to enforce these transfers. 2

4 To overcome this problem, we consider a special type of contract that can provide insurance against reclassification risk but does not require consumers commitment, incomebased transfers or any other special arrangements. These are guaranteed renewable contracts discussed in details by Pauly et al (1995). These contracts are front-loaded: a consumer is required to prepay part of his future premiums and this prepayment locks him into the contract. In return, a consumer is guaranteed that i) he will be able to renew his health insurance contract in the future; ii) the prespecified renewal price will be independent of his future health realizations. A key feature of this contract is that reclassification risk is insured not by making healthy people pay for the sick but by allowing individuals to make state-contingent savings that pay off when their premiums increase. 3 To evaluate welfare costs of reclassification risk, we consider how much welfare improvement can be achieved from introducing guaranteed renewable contracts in the individual health insurance market. 4 We construct a general equilibrium overlapping generations model where people face uninsurable labor income risk and medical expense risk that can be partially insured. Several types of health insurance are available. First, some individuals have access to employer-based insurance. Second, lowest-income individuals can get Medicaid. Finally, all individuals can buy insurance policy directly in the individual market. In the individual market premiums are risk-rated, i.e. depend on the current health conditions of individuals. All policies last for one year while medical expenditures are persistent, which creates the problem of reclassification risk. Our model reflects two institutional features that are important when evaluating the importance of reclassification risk in the U.S. health insurance markets. First, a large 3 It is important to point out the fundamental difference between these guaranteed renewable contracts and the regulatory guaranteed renewability provision that exists in some states of the US. The guaranteed renewability provision forbids insurance companies to deny coverage to individuals who already have an insurance contract and want to renew it. However, this provision does not require that the price at renewal is fixed at some prespecified level. In contrast, the key feature of guaranteed renewable contracts is that they allow to predetermine the renewal price in advance. 4 More specifically, in order to measure welfare costs of reclassification risk we introduce a frictionless market offering guaranteed renewable contracts. An alternative modeling strategy is to allow this market to exist in the baseline economy but with frictions that prevent people from using it. Then we can quantify how large are these frictions. Our results are robust to this alternative modeling strategy. In Appendix D we show that relatively small fixed costs are enough to deter people from buying guaranteed renewable contracts, which possibly explain why we do not observe these contracts offered in reality. 3

5 fraction of non-elderly adults gets their insurance from employer-based market. This market is community rated, i.e. premiums are independent of the health conditions of individuals. People with permanent access to this market are protected from the risk of premium fluctuations. Also, lowest-income individuals can get public insurance from Medicaid for free. Second, for people who face high medical shock and/or bad labor income shock, the government provides protection in the form of the consumption minimum floor. This consumption floor can also mitigate the consequences of the lack of an explicit insurance against reclassification risk. We calibrate the model using the Medical Expenditure Panel Survey dataset to match the key insurance statistics for the U.S. Using the calibrated model we study the quantitative implications of introducing frictionless guaranteed renewable contracts in the individual market. We find that comparing to the situation when only standard short-term insurance contracts are available, introduction of guaranteed renewable contracts can noticeably decrease uninsurance rates - from 25.9% to 19.4% due to the higher participation in the individual insurance market. Also, if both standard and guaranteed renewable contracts are available, most of the consumers prefer to buy the later type of contract. Our results show that people who hold guaranteed renewable contracts face almost no fluctuations in their health insurance premiums even if their health deteriorates. This implies that these contracts provide a good protection against reclassification risk. In terms of welfare, we find that introduction of guaranteed renewable contracts brings small welfare gains. This suggests that in the current U.S. health insurance system people are not very concerned about the absence of an explicit insurance against reclassification risk. This happens because two institutional features provide good implicit insurance against reclassification risk. First, employer-sponsored health insurance that protects mostly high-income people; and second, the consumption minimum floor that protects mostly people with low income. If these two institutional features are removed, the average welfare gains from having access to guaranteed renewable contracts are large and can exceed 2% of the annual consumption. Our results are robust to the alternative 4

6 design of guaranteed renewable contracts and the degree of actuarial unfairness in the health insurance market. This paper is structured as follows. Section 2 reviews the related literature. Section 3 illustrates how a guaranteed renewable contract works using a simple example. Section 4 presents the model. Section 5 explains our calibration. Section 6 discusses the results. Section 7 considers implications of our results for the upcoming health insurance reform. Section 8 concludes. 2 Related literature This paper belongs to two strands of literature. First is the literature studying how private markets can provide insurance against reclassification risk if buyers cannot commit to a contract. A seminal paper in this area is Cochrane (1995) who characterizes a set of contracts that can provide long-term health insurance in such an environment. His insight is to combine standard one-period insurance contracts with premium insurance, i.e. insurance against future premium fluctuations. One requirement for such premium insurance to work is that each consumer needs to open a special account that works as a clearing house between him and the insurance company. An important condition is that consumers cannot freely withdraw money from this account. One special case in this set of contracts that can work without a special account are front-loaded guaranteed renewable contracts. These contracts were studied in more details by Pauly et al (1995) who showed that guaranteed renewable contracts can provide a good degree of reclassification risk insurance without creating liquidity problem if consumers buy them while still young and healthy. Front-loaded contracts were also studied by Hendel and Lizzeri (23) for the case of life insurance market. They showed that the structure of premiums in this market is consistent with front-loaded contracts that emerge in the absence of consumer commitment. However, Fang and Kung (21) and Daily et al (28) showed that the growing life settlement market can limit the degree of reclassification risk insurance that life insurers can provide. Finkelstein et al. (25) studied front-loaded contracts in the 5

7 long-term care insurance market and showed that the amount of front-loading currently existing is not enough to lock consumers into the contracts. To our knowledge, our paper is the first one that studies welfare effects of guaranteed renewable contracts in the health insurance market in a general equilibrium framework and taking into account the existing institutions in the US. The second strand of literature this paper belongs to studies quantitative heterogeneous agent models with incomplete markets augmented by (i) medical expense shocks and (ii) health insurance markets where individuals can partially insure these shocks. This branch of incomplete market literature has emerged recently and includes, among others, papers by Kitao and Jeske (29) who study subsidies for employer-based insurance, Hansen et al (212) who evaluate the consequences of expanding Medicare program, Hsu (29) who studies the effect of private health insurance on savings, and Pashchenko and Porapakkarm (213) who study the current health reform in the U.S. These studies consider an environment when only one type of contract is available in the individual health insurance market. Our contribution to this literature is that we expand the contract space by allowing insurance firms to offer guaranteed renewable contracts. Since it is a long-term contract, this extension involves solving a dynamic contracting problem within a general equilibrium framework. 3 Simple illustration This section constructs a simple example that illustrates how a guaranteed renewable contract works. Consider an individual whose health is good, and the price he pays for a standard one-period health insurance contract is p L. With probability v an individual may still be in good health in the next period, in which case his health insurance premium will stay unchanged. However, with probability 1 v his health status may deteriorate. If this happens, his health insurance premium for the standard contract will raise to p H, where p H > p L. If an individual buys the standard one-period contract, he is exposed to reclassification risk - the risk that his health premium will rise from p L to p H. 6

8 Suppose an individual has an option to buy a guaranteed renewable contract at the price p GR 1. This contract insures his medical expenditure in the next period like the standard one-period contract. On top of that, it guarantees that in the next period an individual can buy health insurance at the prespecified price p GR 2 that does not depend on his health status realization. If his health status remains the same he can buy a standard contract at price p L. However, if his health status deteriorates he can renew his guaranteed renewable contract at price p GR 2 < p H. Under the assumption of perfect competition in the insurance market, the price of such a guaranteed renewable contract is determined in the following way: p GR 1 = p L + (1 v)(p H p GR 2 ). (1) Note that the guaranteed renewable contract is more expensive than the regular oneperiod contract because of the front-loading part (1 v)(p H p GR 2 ). This front-loading takes into account the fact that an individual can become unhealthy but the price of renewing his health insurance (p GR 2 ) cannot be readjusted. 4 Model 4.1 Households Demographics and preferences The economy is populated by two overlapping generations: young and old. A young individual stays young with probability ζ y and becomes old with a probability 1 ζ y. An old individual survives to the next period with probability ζ o. 5 The population is assumed to remain constant. Old agents who die are replaced by the entry of new young 5 We assume a stochastic aging environment because a full life-cycle model is computationally impractical in our framework. The most time-consuming part of our computations is to find equilibrium prices of guaranteed renewable contracts. In a stochastic aging model this price depends only on health status. In the full life-cycle model the price will be a function of both age and health which makes our model computationally infeasible especially when it comes to transition calculations. In Section 5.3 we explain how we adjust our calibration strategy to approximate for the key life-cycle features in our model. 7

9 agents. An individual discounts his future utility by the discount factor β. Preferences are described by the CRRA utility function with the risk aversion parameter σ: u(c) = c1 σ 1 σ. Health insurance An individual s health status h is indexed by {1, 2,.., H}. An increasing number implies deteriorating health status. Health status evolves according to a H-state Markov process, where G y (h h) stands for the young and G o (h h) for the old. The current health status of an individual determines his current medical expenditures x (h), where x is a deterministic and strictly monotone-increasing function, different between the young and the old. 6 Thus, in the following, we will refer to health status (h) and medical expenditures (x) interchangeably. Each young individual can buy insurance against medical expenditures in the individual insurance market where two types of contracts are offered. The first type is a standard one-year contract that covers some fraction of the next period medical expenditures. The price of this contract depends on the current health status of an individual and is denoted by p I (h). The second type of contract is guaranteed renewable. This contract covers a fraction of the next period s medical expenditures like a standard oneyear contract. In addition, a guaranteed renewable contract provides an option to renew insurance in the following period at the same price regardless of the new health status 7. Guaranteed renewable contracts do not have a termination date, i.e. an individual can renew the same contract as long as he is still young. An important condition for an individual to be able to renew this contract is continuous participation. In other words, if an individual does not renew the contract once, he will loose the option to renew it in the 6 We represent medical expenses as exogenous shocks as opposed to an outcome of people s optimal decision making. We discuss this assumption in details in Appendix B. 7 There are several ways to design a guaranteed renewable contract by changing the price that an insurer guarantees at the renewal. In our main experiments we assume that the renewal price is the same as the price of the original contract. Later on we relax this assumption by letting the renewal price to differ from the original price. Detailed discussion of these experiments is provided in section 6. 8

10 future. The premium of a newly issued guaranteed renewable contract is a function of the current health status of an individual. The price of a guaranteed renewable contract that is already in force is fixed and determined by the health status of an individual at the time of the contract initiation. In each period, with some probability, a young individual can get an offer to buy employer-sponsored health insurance (ESHI). This is denoted by g: g = 1 if an individual gets an ESHI offer, g = if he does not. 8 The out-of-pocket premium of employer-based insurance is equal to p = (1 ψ) p. Here p is the premium charged to all participants of the employer-based pool, and ψ is the fraction of this premium paid by the employer. Low-income individuals are eligible to enroll in Medicaid that provides health insurance for free. To become eligible for Medicaid, an individual s total resources net of out-of-pocket medical expenses must be below a certain level which is denoted by y pub. We use i to index the current health insurance status as follows: 2 ; if uninsured 1 ; if insured by Mediciad i = ; if holding a standard one-period insurance or ESHI 1, 2,..., H ; if holding a guaranteed renewable contract originated when his health status equals i. If an individual holds a guaranteed renewable contract, i keeps track of the health status when the contract was initiated. For a newly purchased contract i is the current health status h. We denote the premium for a newly issued guaranteed renewable insurance as p GR (h), and the premium for a guaranteed renewable contract that is already in force as p GR (i) for i = {1, 2,..., H}. 8 We incorporate an important feature of the data that many individuals do not have a permanent access to ESHI but can lose this access several times over their working life. We do not explicitly model COBRA which provides a possibility for some individuals to extend their access to the employer-based pool for a limited time period after they lose their ESHI offer. Modeling COBRA requires us to add an additional state variable making our computation of the transition impractical. In addition, COBRA is not effective in eliminating the problem of fragmented access to ESHI and thus will not conceptually change our model. 9

11 If a young person is insured, the insurance will cover a fraction q (i, x) of his current medical expenses. This fraction depends on his medical expenditures (x) and the type of insurance he has (i). All retired households are enrolled in Medicare. Medicare charges a premium of p med. We denote the fraction of medical expenses covered by Medicare by q med (x). Labor income A young individual supplies labor inelastically. We denote his earnings by wz, where w is the adjusted wage per effective labor unit and z is his idiosyncratic productivity. We model the productivity, an ESHI offer, and health status as a joint Markov process. The productivity of the old is set to zero. Taxation and social transfers Each households has to pay income tax T (y). The taxable income y is based on both labor income and capital income. We incorporate two features of the current U.S. tax code related to the taxation of health-related expenses into our definition of y. First, households can tax-exempt their medical expenses in excess of 7.5% of their income. Second, households buying group insurance can subtract the out-of-pocket group premium p from their taxable income. We also assume a social welfare system, T SI. The social welfare system guarantees that a household will have a minimum consumption level at c. This reflects the U.S. public transfer programs such as Medically Needy part of Medicaid, food stamps, Supplemental Security Income (SSI), and transfers to finance uncompensated care. 9 ss. All old individuals are retired. They receive Social Security benefits in the amount 9 This structure reflects an important feature of the US social insurance system - the coexistence of ex-ante and ex-post insurance. Ex-ante insurance is represented in our model by Medicaid. Medicaid enrolles get into the program based on their current total resources and this program covers their next period s medical expenses. In contrast, ex-post insurance that is represented by the consumption floor is available to individuals impoverished by their out-of-pocket medical costs this period. 1

12 Optimization problem Retired individuals The state variables of an old individual include liquid capital (k K =R + {}) and health status (h H = {1, 2,..., H}). The value function of the old can be written as follows: where V o (k, h) = max u (c) + βζ o E t V o (k, h ) (2) c,k s.t. k (1 + r) + ss + T SI = c + ζ o k + x ( 1 q med (x) ) + p med + T (y) (3) T SI = max (, c + x ( 1 q med (x) ) + T (y) + p med ss k (1 + r) ) (4) y = max (, ỹ) (5) ỹ = rk + ss max (, x ( 1 q med (x) ).75(rk + ss) ) (6) Equation (3) is the budget constraint. We assume that there is an actuarially-fair annuity market. Thus each retired individual needs to save only ζ o k instead of k 1. Equation (6) takes into account the tax-deductibility of medical expenses in excess of 7.5% of the total income. 11 Young individuals The state variables for a young individual include liquid capital (k K =R + {}), health status (h H = {1, 2,..., H}), idiosyncratic labor productivity (z Z =R + ), ESHI offer status (g G = {, 1}), and index of health insurance status (i I = { 2, 1,, 1, 2,..., H}). Each period an individual chooses his consumption (c), saving (k ), and health insurance status for the next period ( i H). Depending on one s Medicaid eligibility, ESHI offer and insurance status, he can choose not to buy any insurance (NB), buy a guaranteed renewable contract (BGR), renew the existing guaranteed renewable contract (RGR), buy a standard individual policy (BI), buy a group insurance (BG), or enroll in Medicaid 1 Alternatively, one can assume that the accidental bequests are evenly distributed to all young. Since the distributed amount is small, it will not affect our results. But the computational cost is higher since one needs to wait until the convergence of total bequests to get the invariant distribution. 11 The problem of a newly retired household is slightly different from a retired household since he is still covered by his pre-retirement insurance. The difference lies in the state variables and the out-of-pocket medical expenditure. For the newly retired, the state variables are {k, h, i}; and in the budget constraint x ( 1 q med (x) ) is replaced by x (1 q (i, x)). 11

13 (BM). We summarize the insurance choices as follows. If a household currently has a guaranteed renewable contract, i = {1, 2, 3,.., H}, 12 {BGR, RGR, BI, BG, BM} if g = 1 and eligible for Medicaid i H = {BGR, RGR, BI, BM} {NB, BGR, RGR, BI, BG} {NB, BGR, RGR, BI} if g = and eligible for Medicaid if g = 1 and not eligible for Medicaid if g = and not eligible for Medicaid If a household does not have a guaranteed renewable contract, i = { 2, 1, }, {BGR, BI, BG, BM} if g = 1 and eligible for Medicaid i H = {BGR, BI, BM} {NB, BGR, BI, BG} {NB, BGR, BI} if g = and eligible for Medicaid if g = 1 and not eligible for Medicaid if g = and not eligible for Medicaid The value function of a working-age household can be written as follows: V y (k, h, z, g, i) = max c,k,i H u (c) + βζ y EV y (k, h, z, g, i ) + β(1 ζ y )EV o (k, h, i ) (7) s.t. k (1 + r) + wz + T SI = c + k + x (1 q (i, x)) + P ( h, i, i H) + T (y) (8) where { } w ; if g = w = w c E ; if g = 1 ; if i H = NB or BM P ( p I (h) ; if i H = BI h, i, i H) = p GR (h) ; if i H = BGR p GR (i) ; if i H = RGR p ; if i H = BG (9) (1) ỹ = { y = max (, ỹ) (11) } wz + rk max (, x (1 q (i, x)).75 ( wz + rk)) ; if i H BG wz + rk max (, x (1 q (i, x)).75 ( wz + rk)) p ; if i H = BG (12) T SI = max (, c + x (1 q (i, x)) + T (y) wz k (1 + r)) (13) 12 Note, that if a household is eligible for Medicaid he cannot stay uninsured because Medicaid is free. 12

14 i = 2 ; if i H = NB 1 ; if i H = BM ; if i H = {BI, BG} i ; if i H = RGR h ; if i H = BGR (14) The conditional expectation on the right-hand side of equation (7) is over {h, z, g }. The second equation is the budget constraint. In equation (9), w is the wage per effective labor unit. If a household has an ESHI offer, then the employer partly pays for the premium. In order to break even, the employer deducts c E from the wage per effective labor unit to get an adjusted wage w. Equation (12) reflects the tax deductibility of the ESHI premium and medical expenses exceeding 7.5% of the income. Equation (14) maps the current health insurance status and health insurance choices into the next period health insurance status. The income eligibility of Medicaid program requires that k (1 + r) + wz x (1 q (i, x)) y pub. Distribution of households To simplify the notations, we denote the space of a household state variables by S: S K H Z G I for young individuals, S K H I for just-retired individuals, and S K H for retirees. Let s S and denote by Γ y (s) and Γ o (s) the measure of young and retired people correspondingly. 4.2 Production sector There are two stand-in firms that act competitively. Their production functions are Cobb-Douglas, AK α L 1 α, where K and L are aggregate capital and aggregate labor and A is the total factor productivity. The first stand-in firm offers ESHI to its workers. The second stand-in firm does not 13. Under the competitive market assumption, the second firm pays each employee his marginal product of labor. Because capital is freely allocated between the two firms, the Cobb-Douglas production function implies that the 13 An alternative setup is that there are two islands, one offers ESHI and the other does not. Workers are stochastically allocated between the two islands but there are no frictions in the capital market. Inside each island, the labor market is competitive. 13

15 capital-labor ratios of both firms are the same. Consequently we have 14 r = αak α 1 L 1 α δ, (15) w = (1 α)ak α L α (16) where δ is the depreciation rate. The first firm has to partially finance health insurance premiums for its employees. These costs are fully passed on to the employees through a wage reduction. In specifying this wage reduction we follow Jeske and Kitao (29). The first firm subtracts an amount of c E from the marginal product per effective labor. The total wage reduction of each employee with an ESHI offer is c E z. 15 The zero profit condition implies c E = ψp ( 1 {i H =BG}Γ y (s) ). (17) 1{g=1} zγ y (s) where 1 { } is a function that is equal to one if its argument is true, otherwise the function is equal to zero. 4.3 Private health insurance sector We model the health insurance sector under the following assumptions. First, both individual and group insurance markets are competitive implying zero expected profit for each insurance contract. Second, there are administrative costs associated with issuing an insurance policy and these costs are proportional to the total value of the contract. Third, 14 Define {K 1, L 1 } and {K 2, L 2 } as aggregate capital and labor in firms 1 and 2. Since capital can ( ) α 1 ( ) α 1 move freely between firms, the Cobb-Douglas production implies r + δ = αa K1 L 1 = αa K2 L 2. Next we can write K L = K 1 + K 2 L 1 + L 2 = K 1 L 1 + K2 L 2 L 2 L L = K 1. 2 L L 1 1 The last equality uses the fact that K1 L 1 = K2 L The assumed structure implies a proportional transfer from high-income to low-income people inside the employer-based pool. This assumption is not important for our results since all changes in our study happen in the individual insurance market. An alternative assumption is a lump-sum wage reduction. This alternative structure is difficult to implement in our setup since some workers will end up earning zero or negative wage. 14

16 health insurance companies can observe only the current health status of an individual. 16 Standard one-period insurance The zero profit condition implies that the premium for a standard one-period insurance contract is equal to the expected discounted medical costs covered by an insurance company multiplied by administrative load ( γ I) : p I (h) = (1 + r) 1 γ I EM (h) (18) Here EM (h) is the expected medical expenses of an individual with health status h covered by an insurance company: EM (h) = h x (h ) q (, x (h )) G y (h h) Guaranteed renewable insurance The price of a newly issued guaranteed renewable contract depends on the current health status of an individual. To determine the premium, an insurer needs to assign a probability to an event that an individual will continue to renew the contract. Consider an individual with health status h t who chooses to buy a new guaranteed renewable contract in period t. Denote by π t+j (h t+j h t ) an insurer s belief that this individual will continue to renew the same insurance contract every period up to a period t + j when his health status becomes h t+j. The zero profit condition allows us to write the premium of a new guaranteed renewable contract as follows: p GR (h t ) = p I (h t ) + 1 (1 + r) j j=1 H h t+j =1 π t+j (h t+j h t ) ( p I (h t+j ) p GR (h t ) ) (19) 16 For standard one-period insurance contracts only health status matters for pricing. For guaranteed renewable contracts an additional factor that affects pricing is the probability that the contract will be renewed in the future. This probability depends not only on health, but also on other state variables, in particular assets and labor income. We do not allow prices to be conditioned on assets or labor income because these variables are difficult for insurance companies to verify. 15

17 The first term on the right hand side is the premium for a standard insurance contract that covers medical expenses in the next period. The second term is the extra payment for the option to renew the contract in the future. It arises because an insurance company will not be able to readjust the price in the future even if an individual s health deteriorates. The beliefs of the insurer π t+j (h t+j h t ) should be consistent with households optimal decisions in equilibrium. Denote the measure of young people with health status h t who ) choose to buy a new guaranteed renewable contract in period t by Γ (h y t, i H t = BGR. ( ) Denote by F h t+j, i H t+j = RGR h t, i H t = BGR the measure of those people in this group who have been renewing the same contract every period from period t to period t + j when their health become h t+j. Thus π t+j (h t+j h t ) can be defined as π t+j (h t+j h t ) = F ( ) h t+j, i H t+j = RGR h t, i H t = BGR Γ ( y h t, i H t = BGR ) (2) Employer-based group insurance The premium in the group insurance market does not depend on the health status of individuals 17. Using the zero profit condition, the premium can be written as a weighted average of the expected covered medical costs of participating employees multiplied by the administrative load ( γ G). p = (1 + r) 1 γ G 1{i H =BG} EM (h) Γ y (s), (21) 1{i H =BG}Γ y (s) 4.4 Government constraint We assume that the government runs a balanced budget. This implies: T (y) Γ y (s)+ (ss T (y) Γ o (s) = + xq med (x) p med) Γ o (s)+ T SI Γ y (s)+ T SI Γ o (s) (22) 17 The U.S. regulation prohibits employers to charge employees with different health-related characteristics different insurance premiums. 16

18 The left-hand side is the total income tax. The first term on the right-hand side is the net expenditure on Social Security and Medicare systems for the old. The last two terms are the costs of running the means-tested transfer program, i.e. to keep households above the consumption minimum floor. We define the competitive equilibrium of the economy in Appendix A. 5 Data and Calibration 5.1 Data We calibrated the model using the Medical Expenditure Panel Survey (MEPS) dataset. The MEPS collects detailed records on demographics, income, medical costs and insurance for a nationally representative sample of households. It consists of two-year overlapping panels and covers the period of We use eight waves of the MEPS, from 1999 to The MEPS links people into one household based on eligibility for coverage under a typical family insurance plan. This Health Insurance Eligibility Unit (HIEU) defined in the MEPS dataset corresponds to our definition of a household. All statistics we use were computed for the head of the HIEU, i.e. we use individual-level data where each individual is a head of a household. We define the head as the person who has the highest income in the HIEU. A different definition of the head (based on gender) does not give significantly different results. We use longitudinal weights provided in the MEPS to compute all the statistics. Given that all individuals are observed for at most two years, we pool together all eight waves of the MEPS. Since each wave is a representation of the population in each year, the weight of each individual was divided by eight in the pooled sample. In our sample we include all non-student heads whose age is at least 2 and whose labor income (to be defined later) is non-negative. The sample size for each wave is 18 We do not use the first two waves of the MEPS because they do not contain the variables we use for constructing a household unit. 17

19 presented in Table 1. We use 23 as a base year. All level variables were normalized to the base year using the Consumer Price Index (CPI). Insert Table 1 here When measuring the insurance status in the data, we use the following approach. In the MEPS the question about the source of insurance coverage is asked retrospectively for each month of the year. We define a person as having employer-based insurance if he reports having ESHI for at least eight months during the year (variables PEGJA- PEGDE). The same criteria was used when defining public insurance (variables PUBJA- PUBDE) and individual insurance status (variables PRIJA-PRIDE) 19. In addition, we assume that a person has an ESHI offer if he reports having an offer in at least two out of three interview rounds during a year (variables OFFER31x, OFFER42x, OFFER53x). 5.2 Demographics, preferences and technology The period in the model is one year. Young agents are born at age 2 and stay young on average 45 years, so the probability to stay young, ζ y, is set to 44/45. The survival probability of an old individual ζ o is set to make the fraction of the old in the population equal to 2%; thus 1 ζ o = 4 (1 ζ y ). To keep the total measure of population equal to one, the measure of newborns in every period is set to (1 ζy ) (1 ζ o ) 2 ζ y ζ o. The risk aversion parameter σ is equal to 3 which is in the range commonly used in the macroeconomic literature. The discount factor β is calibrated to match the aggregate capital output ratio of 3.. The Cobb-Douglas function parameter α is set to.33 which corresponds to the U.S. s capital income share. The annual depreciation rate δ is calibrated to achieve the interest 19 For those few individuals who switch the source of coverage during the year, we define insurance status in the following way. If a person has both ESHI and individual insurance in one year, and each coverage lasted for less than eight months but with a total duration of coverage of more than eight months, we classify this person as individually insured. Likewise, when a person has a combination of individual and public coverage that altogether lasts for more than eight months, we define that individual as having public insurance. Our results do not change significantly if we change the cutoff point to 6 or 12 months. 18

20 rate of 4.% in the baseline economy. The total factor productivity A is normalized to make the average labor income equal to one in the baseline model. 5.3 Joint process of health, labor income, and ESHI offer Health status and Medical expenses The medical expenses in our model correspond to the total amount paid for the health care services (variable: TOTEXP). This includes both out-of-pocket payments and payments made by insurance companies but it does not include over-the-counter drugs. In our model there is a one-to-one mapping between medical expenses and health status. We categorize medical expenses into five bins and each bin corresponds to a different health status (Table 2). Insert Table 2 here The average amount of medical expenses corresponding to each health status are (.1,.16,.75,.318,1.483) for young households and (.21,.83,.251,.917,2.317) for retired households. These numbers are based on the medical expenses in 23/24 wave normalized by the average labor income ($35, 624). To construct a transition matrix for health status, we compute the fraction of household moving from one bin to another. 2 The resulting transition matrix for young households, G y (h h), is , We assume that medical shocks follow a Markov process since MEPS allows us to observe only two consecutive periods of medical expenses for each individual. The implied first and second autocorrelation of total medical expenses are.37 and.16. French and Jones (24) provide a detailed examination of health costs in Health and Retirement (HRS) dataset and report first and second autocorrelation equal to.45 and.34 correspondingly. It is important to point out that these numbers should be compared with caution for the following reasons. First, French and Jones s estimates are based on out-of-pocket medical expenses while ours - on total medical expenses. Second, HRS includes only people above age 5 while our estimates include young people and medical shocks for young people are less persistent. Finally, data frequency for medical expenses in MEPS is one year while in HRS it is two years. 19

21 while the transition matrix for retired households, G o (h h), is Here the first row corresponds to h = 1 and the first column corresponds to h = 1. Labor income We define labor income as a sum of wages (variable WAGEP) and 75% of income from business (variable BUSNP). This definition is the same as used in the Panel Study of Income Dynamics Dataset (PSID) that has been commonly used for income calibration in the macroeconomic literature. We categorized labor income into five quintiles (5 2%). The labor income level in each quintile is based on the value for 23/24 wave normalized by the average income. These numbers are.91,.477,.82, 1.226, and The dashed lines in Panels (a) and (b) in Figure 1 show the relationship between labor income and medical expenses/health observed in the data. The hump shape in Panel (a) can be explained by the life-cycle profile of labor income. Our model does not have age dimension so the age profile of labor income is partially captured by health status. In the data households in good health (h = 1) are more likely to be young, while those in bad health (h = 4 or h = 5) are more likely to be near retirement. These two groups tend to have lower incomes than the middle-age households. Panel (b) also shows that the average medical expenses of households in the first income quintile are two times higher than the average medical expenses of the high income group. This pattern is driven by two facts. First, the distribution of medical expenses is highly skewed: the medical expenses of people with h = 5 is more than four times higher than the medical expenses of those with h = 4. Second, households with serious health problems, h = 5, are more likely to experience a very low income shock. 2

22 Insert Figure 1 here When constructing a joint Markov process of labor income and health status, our goal is to capture the above pattern. To do this we divide our sample into four subsamples based on the health status in the second year of each wave. The first, second, and third subsamples include households whose health status in the second year equals 1, 2, and 3 respectively. The forth subsample include households whose health status in the second year equals 4 or 5. Then we construct a transition matrix of labor income for each subsample by calculating the fraction of households who move from one quintile to another. The resulting four transition matrixes capture the dynamics of labor income conditional on health shock in the second period, and are denoted as Q (z z, h = 1), Q (z z, h = 2), Q (z z, h = 3), and Q (z z, h = 4). Due to the small sample size, we cannot get the transition matrix conditional on h = 5 directly. So we define Q (z z, h = 5) = a Q (z z, h = 4) + (1 a) D; a 1, where D is a 5 5 matrix with the first column equal to one and the remaining columns equal to zero. If a = 1, Q (z z, h = 5) = Q (z z, h = 4). But if a =, Q (z z, h = 5) = D, meaning that the income of those households who have serious health problems drops to the level of the lowest income quintile. In our calibration, we choose a to make the average labor income of those with h = 5 match the data as shown in Panel (a) of Figure 1. The joint transition matrix of health status and labor income is constructed by combining the transition matrix of health status, G y (h h), with the conditional transition matrix of labor income Q (z z, h ). The advantage of this approach is that the conditional expected medical expenses depend only on the current health status. This dramatically simplifies the computation since we can compute the premiums of standard one-period insurance directly from G y (h h) If the conditional expected medical expense also depend on the current labor income, say E (x x, z = 1) E (x x, z = 2), and the insurance company does not observe z, then the premiums of standard one-period contracts will depend on households insurance decision and the equilibrium distribution of households. 21

23 ESHI Offer status The dashed line in Panel (d) in Figure 1 shows that there is a strong correlation between the probability to get access to ESHI and labor income. We assume that the probability of getting an ESHI offer is a logistic function: P rob t = exp(u t) 1 + exp(u t ), where the variable u t is an odds ratio that takes the following form: u t = η + η 1 D gt 1 + η h D ht + η z D zt + η year D year, (23) where D gt 1 is a dummy variable for an ESHI offer in period t 1, D ht and D zt are the sets of dummy variables for health status and income quintile in period t, and D year is a set of dummy variables for each year. To calibrate the joint distribution {h, z, g} of newborns, we use the empirical joint distribution of households aged 2-35 from the data. This allows us to approximate the life-cycle features absent from our model. In particular, an important characteristic of a life-cycle model is an increasing age profile of medical expenses and labor income. In our calibration newborns are clustered around relatively low medical expenses and relatively low labor productivity. Given the persistence of the processes for medical shocks and productivity, young people in our model are slowly moving up the medical expenses and labor income ladders. The average medical expense as a fraction of average labor income in our model is 7.1% for young people and 2.% for old people, comparing to 7.3% and 2.4% in MEPS. For newborns, the fraction of average medical expense in average labor income constitutes 4.6% comparing to 5.2% for people aged 2-35 in MEPS. Figure 1 allows to compare our simulations of {h, z, g} with the data (simulations are plotted with the solid lines). Overall, we are able to match the key features of the data well. However, the simulated offer rate (59.1%) is slightly lower than in the data (64%) This mismatch mostly arises from the absence of educational heterogeneity in our model. As shown in Pashchenko and Porapakkarm (211), people with low educational attainment have a significantly 22

24 5.4 Insurance policies We use the MEPS to find the fraction of medical costs covered by an average insurance policy. We estimate the following equation InsCov = β + β 1 x + β 2 x 2 + ΘD year separately for private insurance, Medicaid, and Medicare. InsCov is medical expenses paid by insurance (variables: TOTPRV,TOTMCD,TOTMCR). We include only people with positive medical expenses when estimating this regression. Then we use our estimates to compute the fraction of medical expenses covered by insurance for each health status and truncate it to be between and 1. Table 3 reports the results for each type of insurance. Insert Table 3 here 5.5 Government constraint In calibrating the tax function T (y) we use a nonlinear relationship specified and estimated by Gouveia and Strauss (1994): T (y) = a [ y (y a 1 + a 2 ) 1/a 1 ] Here a controls the marginal tax rate levied on people with the highest income, a 1 determines the progressivity of the tax code, and a 2 is a scaling parameter. We set a and a 1 to the original estimates of Gouveia and Strauss (.258 and.768 correspondingly). The parameter a 2 is used to balance the government budget. The consumption minimum floor c in the baseline economy was calibrated so that the fraction of households with assets less than $5, in the model is the same as in the data. Based on the Survey of Consumer Finance (SCF) dataset this fraction is lower probability to get access to ESHI. 23

25 2.% (Kennickell, 23). To match this fraction, c is set to.92 of the Federal Poverty Line (FPL), or $8, 87. The Social Security replacement rate is set to 45% of the average labor income. This number is obtained by applying the Social Security benefit formula to the average labor earnings profile. 5.6 Medicaid and private insurance The Medicaid eligibility rules differ from state to state. As of 29, 14 states had an income eligibility threshold below 5% of FPL, 2 states had it between 5% and 99% of FPL, and 17 states had it higher than 1% of FPL (Kaiser Family Foundation, 28). We set y pub to 48.% of FPL, or $4, 595, to match the fraction of people insured by Medicaid. In our baseline model, we assume that only standard one-year contracts are offered in the individual market. To match the fraction of those buying individual insurance, we set the administrative load of an individual insurance policy γ I to The administrative load for the group insurance γ G is set to 1.11 (Kahn et al, 25). We set the share of health insurance premium paid by the firm (ψ) to 83.%. This number is consistent with the data in which the premiums of group insurance paid by employers range from 77% to 89% (Sommers, 22). 5.7 Performance of the baseline model Tables 4 and 5 summarize the parameters used in our baseline model. Table 6 reports the fraction of non-elderly adults with different insurance statuses and the numerical results from the baseline model. The model slightly underestimates the fraction of people with ESHI because our calibrated offer rate is lower than that in the data. As a result the fraction of uninsured is slightly overestimated. Insert Table 4, Table 5, and Table 6 here 24